Multifractal analysis of measures arising from random substitutions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929690215710720 |
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| author | Mitchell, Andrew Rutar, Alex |
| author_facet | Mitchell, Andrew Rutar, Alex |
| contents | We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the $L^q$-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the $L^q$-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word $L^q$-spectrum of a random substitution and show that this coincides with the $L^q$-spectrum of the corresponding frequency measure for all $q \geq 0$. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multifractal analysis of measures arising from random substitutions Mitchell, Andrew Rutar, Alex Dynamical Systems 37B10, 37C45, 52C23 We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the $L^q$-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the $L^q$-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word $L^q$-spectrum of a random substitution and show that this coincides with the $L^q$-spectrum of the corresponding frequency measure for all $q \geq 0$. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy. |
| title | Multifractal analysis of measures arising from random substitutions |
| topic | Dynamical Systems 37B10, 37C45, 52C23 |
| url | https://arxiv.org/abs/2301.04958 |